Systems, methods and computer program products for constructing sampling plans for items that are manufactured

ABSTRACT

A desired Acceptable Quality Limit (AQL), a desired Key Defect Rate (KDR), a desired power of a sampling plan for items that are manufactured and a desired false alarm rate for the sampling plan are input into a computer. The computer calculates a required sample size to provide the desired AQL, the desired KDR, the desired power of the sampling plan for the items that are manufactured and the desired false alarm rate for the sampling plan. Thus, each of the individual parameters may be independently specified based on the items that are manufactured, desired AQLs, KDRs, power and false alarm rates. Reliance on ANSI/ASQ Z1.9 tables which might best fit a user&#39;s desired parameters can be reduced and preferably eliminated. In addition to calculating the required sample size, a decision rule critical value also may be calculated based upon the required sample size to provide the desired AQL, the desired KDR, the desired power and the desired false alarm rate for the sampling plan. Following the calculations, a relationship between sample size, acceptable number of defective items and false alarm rate automatically may be graphically displayed based upon the desired AQL, the desired KDR and the desired power of the sampling plan. The items that are manufactured may then be sampled at the required sample size to obtain samples, and the number of defective items in the samples or other response variables in each of the samples, may be measured. After measuring the response variables, such as the number of defective items, the measured response variable for each of the samples is input into the computer and an estimate of the Quality Level (QL) for the items that are manufactured is calculated, based on the measured response variable for each of the samples.

RELATED APPLICATIONS

[0001] This application is a divisional of pending application Ser No.09/397,357 filed on Sep. 15, 1999.

FIELD OF THE INVENTION

[0002] This invention relates to data processing systems, methods andcomputer program products, and more particularly to statistical systems,methods and computer program products.

BACKGROUND OF THE INVENTION

[0003] Sampling plans are widely used in manufacturing environments inorder to determine whether items are being manufactured at a desiredquality level. In order to construct a sampling plan, the ANSI/ASQ Z1.9standard generally is used. The ANSI/ASQ Z1.9 standard is a collectionof sampling plans presented in tabular and graphical form. In order toconstruct a sampling plan, the ANSI/ASQ Z1.9 standard is studied, and asampling plan which best matches a desired sampling plan is selected.

[0004] Unfortunately, in using the ANSI/ASQ Z1.9 standard, the user maybe bound to those sampling plans that are set forth in the standard. Inparticular, specific values of error rates, power, sample size and otherparameters may be forced upon a user because the tables may not includethe exact criteria that are desired by a given user.

[0005] Moreover, because the ANSI/ASQ Z1.9 standard uses test proceduresthat are based on a non-central t distribution, it may be difficult forthe user to interpolate or extrapolate between tables of the standard.Notwithstanding these difficulties, the ANSI/ASQ Z1.9 standard continuesto be widely used in constructing sampling plans for items that aremanufactured.

SUMMARY OF THE INVENTION

[0006] It is therefore an object of the present invention to provideimproved systems, methods and computer program products for constructingsampling plans for items that are manufactured.

[0007] It is another object of the present invention to provide systems,methods and computer program products for constructing sampling plansthat can be flexible to meet the needs of a particular user andmanufacturing process.

[0008] These and other objects are provided, according to the presentinvention, by inputting into a computer a desired Acceptable QualityLimit (AQL), a desired Key Defect Rate (KDR), a desired power of asampling plan for the items that are manufactured and a desired falsealarm rate for the sampling plan. The computer then calculates arequired sample size to provide the desired AQL, the desired KDR, thedesired power of the sampling plan for the items that are manufacturedand the desired false alarm rate for the sampling plan. Thus, each ofthe individual parameters may be independently specified based on theitems that are manufactured, desired AQLs, KDRs, power and false alarmrates. Reliance on ANSI/ASQ Z1.9 tables which might best fit a user'sdesired parameters can be reduced and preferably eliminated.

[0009] In addition to calculating the required sample size, a decisionrule critical value also may be calculated based upon the requiredsample size to provide the desired AQL, the desired KDR, the desiredpower and the desired false alarm rate for the sampling plan. Followingthe calculations, a relationship between sample size, acceptable numberof defective items and false alarm rate automatically may be graphicallydisplayed based upon the desired AQL, the desired KDR and the desiredpower of the sampling plan.

[0010] The items that are manufactured may then be sampled at therequired sample size to obtain samples, and the number of defectiveitems in the samples or other response variables in each of the samples,may be measured. After measuring the response variables, such as thenumber of defective items, the measured response variable for each ofthe samples is input into the computer and an estimate of the QualityLevel (QL) for the items that are manufactured is calculated, based onthe measured response variable for each of the samples.

[0011] Prior to calculating the required sample size and the decisionrule critical value, a sample distribution that is variance invariantmay be calculated based on a normal distribution. A percentile grid ofsample size and a true process defect rate is formulated based onestimated percentiles of a cumulative distribution of the samplingdistribution. A bias-corrected percentile grid of sample size and thetrue process defect rate is then formulated from the percentile grid.The bias-corrected percentile grid is stored in the computer.

[0012] The bias-corrected percentile grid may be used to compute thedecision rule critical value from the AQL and the false alarm rate,across a plurality of sample sizes. The bias-corrected percentile gridis evaluated for the decision rule critical value, to determine therequired sample size. More particularly, the decision rule criticalvalue is computed from the AQL, the false alarm rate and the desiredsample size using the bias-corrected percentile grid of sample size anda true process defect rate. The bias-corrected percentile grid isevaluated for values that are larger than the AQL, with the percentilebeing the desired power.

[0013] After the measured response variable for each of the samples isinput into the computer, an estimate of the QL may be calculated bycomputing a bias correction coefficient. A QL test statistic is computedas a function of the bias correction coefficient and at least onequantile from a cumulative distribution function of a central tdistribution, with at least one argument that is a function of a samplemean, a sample standard deviation, the sample size and a specificationlimit. The computer automatically can determine whether the QL teststatistic is at least equal to the decision rule critical value.

[0014] In another aspect of the invention, after the measured responsevariables are input to the computer, the computer calculates a pointestimate of the number of out-of-specification items that aremanufactured based on the measured response variable for each of thesamples.

[0015] In addition to calculating a required sample size as wasdescribed above, the present invention also may be used to calculate aKDR that is produced from a desired sample size. In particular, adesired sample size, a desired false alarm rate, a desired AQL and adesired power are input into a computer. The computer calculates a KDRthat is produced from the desired sample size, the desired false alarmrate, the desired AQL and the desired power of the sampling plan for theitems that are manufactured. Thus, given a desired sample rate, a KDRmay be calculated.

[0016] The KDR may be calculated by computing a decision rule criticalvalue based on the desired AQL and the desired false alarm rate for thedesired sample size. After the KDR is calculated, a relationship betweenacceptable number of defective items and false alarm rate may begraphically displayed based on the desired AQL, the desired KDR and thedesired power of the sampling plan.

[0017] As described above, after the calculation is made, the items maybe sampled at the desired sample size to obtain samples, and a KDR ofthe items that are manufactured may be determined from the samples.After the items are sampled, the measured response variable for each ofthe samples may be provided to the computer, and the computer cancalculate an estimate of the KDR, so that the estimate of the KDR can becompared to the KDR that was calculated. After providing the computerwith the measured response variable for each of the samples, a pointestimate of a process defect rate for the items that are manufacturedalso may be calculated based on the measured response variable for eachof the samples.

[0018] The present invention may be embodied in one or more spreadsheetswith an efficient user interface. The spreadsheets may be used in lieuof the ANSI/ASQ Z1.9 standard, to allow flexible sampling plans to beconstructed and the results of sampling to be measured without the needto fit the desired sampling plan to conform to one of the series ofcharts in the ANSI/ASQ Z1.9 standard. It will be understood that thepresent invention may be embodied as systems, methods and/or computerprogram products.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019]FIG. 1 is a simplified block diagram of systems, methods andcomputer program products for constructing sampling plans for items thatare manufactured according to the present invention.

[0020]FIG. 2 is a flowchart illustrating operations for calculating arequired sample size according to the present invention.

[0021]FIG. 3 is a flowchart illustrating operations for determining abias-corrected percentile grid according to the present invention.

[0022]FIG. 4 is a flowchart illustrating operations to calculate anestimate of quality level according to the present invention.

[0023]FIG. 5 is a flowchart illustrating operations for calculating aKDR according to the present invention.

[0024] FIGS. 6A-6F graphically illustrate histograms of Monte Carloreplicates that indicate the shape of a probability distributionfunction for {circumflex over (p)}.

[0025]FIG. 7 graphically illustrates bias as a function of sample size,when p=AQL.

[0026]FIGS. 8A and 8B illustrate bias curves across various sample sizesfor an AQL of 0.01 and 0.10, respectively.

[0027]FIG. 9A graphically illustrates bias correction regression forhypothesis testing, for n=5.

[0028]FIG. 9B graphically illustrates bias correction regression for QLestimation, for n=5.

[0029]FIG. 10 is a screen shot of an attributes work sheet according tothe present invention.

[0030]FIG. 11 is a screen shot of the work sheet of FIG. 10, for Example1.

[0031]FIG. 12 is a screen shot of an attributes data initialspreadsheet, for Example 2.

[0032]FIG. 13 is a screen shot of an attributes data final spreadsheet,for Example 2.

[0033]FIG. 14 is a screen shot of a variables spreadsheet according tothe present invention.

[0034]FIG. 15 is a screen shot of a variables data spreadsheet, forExample 3.

[0035]FIG. 16 is a screen shot of a variables data spreadsheet 14, forExample 4.

[0036]FIG. 17 is a screen shot of a variables data analysis spreadsheetaccording to the present invention.

[0037]FIG. 18 is a screen shot of a variables data analysis spreadsheet,for Example 5.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0038] The present invention now will be described more fullyhereinafter with reference to the accompanying drawings, in whichpreferred embodiments of the invention are shown. This invention may,however, be embodied in many different forms and should not be construedas limited to the embodiments set forth herein; rather, theseembodiments are provided so that this disclosure will be thorough andcomplete, and will fully convey the scope of the invention to thoseskilled in the art. Like numbers refer to like elements throughout.

[0039] The present invention can provide a simple, flexible tool to anon-statistical audience that can allow a user to both construct anappropriate variables sampling plan to meet desired needs, as well as toanalyze the data once it is collected. Specifically, the presentinvention can allow the user to prescribe the Acceptable Quality Limit(AQL), Key Defect Rate (KDR), the power of the test, and the level (alsoreferred to as either a type I error or the false alarm rate) of thetest. These four values may then be used to determine the requiredsample size as well as the critical value for the test statistic.

[0040] Moreover, according to the invention, the user can specify asample size, and given the AQL, power and false alarm rate, determinethe resulting KDR. This aspect of the invention can enable the user toevaluate the adequacy of a specific sample size for the particular itemsbeing manufactured.

[0041] The above-described parameters may be calculated, according tothe invention, using a central t distribution and extensive Monte Carlosimulation that can provide the user with a flexible, efficient,non-tabulated tool to monitor and maintain the AQL of a process viaacceptance sampling.

[0042] The sampling plans may be constructed using Microsoft® Excelworkbooks or other spreadsheets, so that special statistical softwareneed not be used and the wide range access of users to Microsoft Officesoftware may be leveraged.

[0043] In the Detailed Description that follows, a General Overview ofthe present invention will first be provided. Then, a DetailedTheoretical Description will be provided. Finally, a DetailedDescription of a User Interface for the present invention will beprovided.

[0044] 1. General Overview

[0045] Referring now to FIG. 1, a simplified block diagram of thepresent invention is shown. As shown in FIG. 1, systems, methods andcomputer program products for constructing a sampling plan for itemsthat are manufactured, according to the invention, may be provided usinga personal computer including a Central Processing Unit (CPU) 100, akeyboard 110, a mouse 120 and/or other user input devices, and a display130 and/or other user output devices. The CPU 100 may include anoperating system 140 and a spreadsheet 150. Sampling plans 160 accordingto the present invention may be included, for example, as spreadsheetworkbooks.

[0046] In a preferred embodiment of the present invention, the CPU 100may be a standard IBM-compatible, Apple or other personal computer, theoperating system may be Microsoft Windows and the spreadsheet 150 may beMicrosoft Excel. However, it will be understood that other generalpurpose computers, including mainframe, midrange, workstation and/orpersonal computers and/or one or more applications running on one ormore of these computers, and/or special purpose hardware may be used.

[0047] Various aspects of the present invention are illustrated indetail in the following figures, including flowchart illustrations. Itwill be understood that each block of the flowchart illustrations, andcombinations of blocks in the flowchart illustrations, can beimplemented by computer program instructions. These computer programinstructions may be provided to a processor or other programmable dataprocessing apparatus to produce a machine, such that the instructionswhich execute on the processor or other programmable data processingapparatus create means for implementing the functions specified in theflowchart block or blocks. These computer program instructions also maybe stored in a computer-readable memory that can direct a processor orother programmable data processing apparatus to function in a particularmanner, such that the instructions stored in the computer-readablememory produce an article of manufacture including instruction meanswhich implement the functions specified in the flowchart block orblocks. Accordingly, blocks of the flowchart illustrations supportcombinations of means for performing the specified functions,combinations of steps for performing the specified functions and programinstruction means for performing the specified functions. It also willbe understood that each block of the flowchart illustrations, andcombinations of blocks in the flowchart illustrations, can beimplemented by special purpose hardware-based computer systems whichperform the specified functions or steps, or by combinations of specialpurpose hardware and computer instructions.

[0048] Prior to providing a general overview of the present invention,an overview-of terms, symbols and abbreviations common to thisstatistical field are listed in Table 1: TABLE 1 Term/Symbol/Abbreviation Meaning Defect Any item that does not meet all relevantacceptance criteria AQL Acceptable quality limit. The numerical valuefor the process defect rate used to calibrate the false alarm rate ofthe sampling plan KDR Key defect rate. The smallest value for theprocess defect rate that will be detected with a preset “highprobability” when using the sampling plan. The numerical value for theprocess defect rate used to calibrate the power of the sampling plan H₀Null hypothesis. The statement being tested. Usually, this claimcontains an “equality”. H₁ Alternate hypothesis. A statement inopposition to the to the null hypothesis. Usually, this is the claimhoped to be proven; hence it is sometimes referred to as the Researchhypothesis. In quality settings, this is usually connected with aprocess shift that must be detected. α Type I error. The probability ofrejecting a lot when the process defect rate equals AQL. Also called thefalse alarm rate. β Type II error. The probability of accepting a lotwhen the process defect rate equals KDR. power The probability ofrejecting a lot when the process defect rate equals the KDR. n Thesample size. p The true, albeit unknown, process defect rate. left-sidedA setting where only a lower specification limit is used. right-sided Asetting where only an upper specification limit is used. two-sided Asetting where both lower and upper specification limits are used. OCcurve Operating Characteristic Curve. A plot of the relationship betweenthe type II error for the test (vertical axis) and the defect rate(horizontal axis). AOQ curve Average outgoing Quality Curve. A plot ofthe relationship between the long-term outgoing defect rate (verticalaxis) and the defect rate (horizontal axis AOQL Average Outgoing QualityLimit. The maximum long-term expected outgoing defect rate. AttributesData X The number of defective items in the sample. c The acceptablenumber of defective items in the sample. Also called the decision rulecritical value. Variables Data {circumflex over (p)} The estimate forthe process defect rate. Also called the quality level, or QL, estimate.k_(L) The lower specification limit for the response variable inoriginal units. k_(U) The upper specification limit for the responsevariable in original units. K The decision rule critical value. ŷ Themean, or average, of the sample data. s The standard deviation of thesample data.

[0049] It will be understood by those having skill in the art thatVariables Data relates to continuous measurements of various parameters,whereas Attributes Data relates to integer values of actual results.

[0050] The present invention may be used to determine a required samplesize and also to determine a KDR. FIG. 2 is a flowchart of operationsfor calculating a required sample size according to the presentinvention. In general, a bias-corrected percentile grid is determinedand input into the computer at Block 210. At Block 220, the desired AQL,KDR, power and false alarm rate are input into the computer. At Block230, a decision rule critical value is computed based upon the requiredsample size to provide the desired AQL, the desired KDR, the desiredpower and the desired false alarm rate. The decision rule critical valuepreferably is determined across a plurality of sample sizes using thebias-corrected percentile grid of sample size and a true process defectrate. Then, the bias-corrected percentile grid is evaluated for thedecision rule critical value to determine the required sample size.

[0051] In particular, the decision rule critical value, K, is computedbased on the numerical values for AQL and α, across several potentialsample size values. For a selected sample size, the bias correctedpercentile grid is used to obtain the numerical value for K, where pequals AQL and the percentile used is 1-α. Interpolation may be usedwhen necessary. Given K, the bias corrected percentile grid is evaluatedwith p=KDR and the percentile equaling the power. This second gridresult should equal K for one particular sample size. Again,interpolation may be used when necessary.

[0052] Continuing with the description of FIG. 2, at Block 250, therelationship between the sample size, acceptable number of defectiveitems and false alarm rate optionally is displayed. Other relationshipsalso may be displayed in a graphical and/or table format. At Block 260,the items are sampled and the response variable is measured. Finally, atBlock 270, the measured response variable for each of the samples isinput into the computer and a calculation is made of an estimate of aquality level (QL) of items that are manufactured, based on the measuredresponse variable for each of the samples.

[0053] Referring now to FIG. 3, details of determining thebias-corrected percentage grid (Block 210 of FIG. 2) will now bedescribed. At Block 310, a sampling distribution that is varianceinvariant is calculated. It is assumed that the response variable ofinterest from a process possesses a normal distribution with mean μ andvariance σ². It also is assumed that the response variable is givencertain specification limits, namely a lower specification limit, k_(L),and/or an upper specification limit, k_(U). The true underlyingproportion, p, of non-conforming (i.e. out-of-specification ordefective) items produced by the process is a function of both μ and σ².Specifically, any of these three parameters can be solved for givennumerical values for the other two parameters. A preliminary Monte Carlostudy concluded that for a fixed value of p, the sampling distributionfor the (left-sided case) statistic${\hat{p} = {P\left( {T^{*} \leq \frac{k_{L} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}} \right)}},{{{with}\quad T^{*}} \sim t_{n - 1}},$

[0054] is invariant to the selection of σ². The right-sided andtwo-sided cases follow analogously.

[0055] The mathematical form of the distribution characterizing{circumflex over (p)} is presently unknown, except by a vague and verycomplex expression. However, Monte Carlo simulation may be used toestimate the percentiles of the cumulative distribution for {circumflexover (p)}, as well as to estimate the expected value for {circumflexover (p)}. Both are functions of the sample size, n, as well as the trueprocess defect rate, p. Therefore, at Block 320, a percentile grid isestablished in three dimensions: (1) the sample size, (2) the trueprocess defect rate, and (3) the percentile to be referenced.

[0056] Unfortunately, the point estimate {circumflex over (p)} is abiased estimate, specifically an overestimate, of the true processdefect rate p. However, {circumflex over (p)} is asymptoticallyunbiased; that is, the bias goes to zero as the sample size gets large.For each value of p, a regression model of the bias as a function of thesample size may be used to determine the multiplication constant neededto eliminate the bias. Thus, at Block 330, a bias corrected percentilegrid is constructed by taking the Monte Carlo based percentile grid andmultiplying each entry by an appropriate constant.

[0057] Referring now to FIG. 4, detailed operations to calculate anestimate of quality level (Block 270 of FIG. 1) will now be described.In particular, a bias-correction coefficient, based on the assumptionthat p=AQL, is computed at Block 410. The QL test statistic is thencomputed at Block 420 as a function of this bias-correction coefficientand quantile(s) from the cumulative distribution function of the centralt-distribution with an argument(s) that is/are a function of the samplemean, sample standard deviation, sample size, and specificationlimit(s). Finally, at Block 430, the decision rule is defined asfollows: reject the hypothesis that the process is operating under thecondition of an AQL defect rate if and only if the QL test statistic isgreater than or equal to the decision rule critical value, K. Otherwise,the process is deemed to be operating under an acceptable defect rate.

[0058] Referring now to FIG. 5, according to another aspect of thepresent invention, the KDR that is produced from a desired sample size,a desired false alarm rate, a desired AQL and a desired power, may becalculated As shown in FIG. 5, a bias-corrected percentile grid may becalculated at Block 210, as was described in connection with FIGS. 2 and3. It will be understood that the calculation of the bias-correctedpercentage grid may only need to be performed once and then may be usedfor all subsequent evaluations and calculations. Then, at Block 510, anacceptable AQL, a desired false alarm rate (α), a desired sample size(n), and a desired power at the unknown KDR are input into the computer.

[0059] At Block 520, the decision rule critical value is computed basedon the desired AQL and the desired false alarm rate for the desiredsample size. In particular, the decision rule critical value, K, basedon the numerical values for AQL, and α is computed for the given samplesize. The bias corrected percentile grid is used to obtain the numericalvalue for K, where p equals AQL and the percentile used is 1-α.Interpolation may be used when necessary. Given K, the bias correctedpercentile grid is now evaluated across several values of p>AQL, withthe percentile equaling the power. This second grid result should equalK for one particular value of p, which becomes the computed value forthe KDR. Again, interpolation may be used when necessary.

[0060] Continuing with the description of FIG. 5, optionally at Block530, a relationship between acceptable number of defective items andfalse alarm rate may be displayed. Alternatively, other relationshipsmay be displayed.

[0061] Then at Block 260, the items are sampled and the responsevariables are measured as was described in connection with FIG. 2.Finally at Block 540, a KDR that is produced from the desired samplesize, the desired false alarm rate, the desired AQL and the desiredpower of the sampling plan is calculated. More particularly, an estimateof the KDR is calculated for the items that are manufactured based onthe measured response variable for each of the samples. Moreover, apoint estimate of a process defect rate for the items that aremanufactured may be calculated based on the measured response variablefor each of the samples. These parameters may be displayed.

[0062] In particular, to obtain a point estimate for the process defectrate, an alteration of the QL test statistic may be performed. This isdue to the form of bias-correction coefficient, which no longer mayutilize the assumption that p=AQL that results from the formalhypothesis test. A family of reference regression models (each definedby the sample size) of the true defect rate versus the expected value of{circumflex over (p)} is constructed based on the Monte Carlo resultsthat produced the original percentile grid. The QL point estimate,{circumflex over (Q)}L, is a transformation of {circumflex over (p)}.For the given sample size, the coefficients from the appropriatereference regression model may be used to transform {circumflex over(p)} into {circumflex over (Q)}L. Interpolation may be used asnecessary.

[0063] 2. Detailed Theoretical Description

[0064] A detailed theoretical description of the statistical theoryunderlying the present invention now will be provided.

[0065] The process parameter of interest is the probability, p, that anindividually produced item is not within specifications. Hence, p isreferred to as the process defect rate. Generally, it is only desirableto protect against an excessively large value for p. The formalhypothesis statement takes the form of a one-sample proportion test,namely

H₀: p=p₀

H₁: p=p₁>p₀

[0066] where p₀ equals the acceptable quality level, or AQL. The falsealarm rate, a, for this test is controlled at a specific level under thecircumstance that p=AQL. Furthermore, the power of this test iscalibrated when the process defect rate equals a value referred to asthe key defect rate, or KDR. Thus, p=KDR, where KDR=p₁>p₀.

[0067] There are two scenarios to be considered when constructing thesampling plan. In both cases the AQL, α, and power are fixed in advance.In the first case, the KDR is specified in advance and the requiredsample size is computed. In the second case, the sample size isspecified in advance with the resulting KDR being computed. In eithercase, it is of interest to estimate the unknown process parameter p fromthe observed (collected) data during the data analysis.

[0068] 2.1 Test Procedure Developmental Theory

[0069] Assume that a lower one-sided specification limit is employed.Let {circumflex over (p)}=P(y≦k_(L)) be a point estimate for p, wherek_(L) represents the lower specification limit and y represents anyfuture observation. Assume that the response variable is normallydistributed with mean μ and variance σ², written as y˜N[μ,σ²], and thatthe observations are all independent of one another. From distributiontheory, with {overscore (y)} being the sample mean and s being thesample standard deviation arising from a sample of n observations:${y - \overset{\_}{y}} \sim {N\left\lbrack {0,{\sigma^{2} + \frac{\sigma^{2}}{n}}} \right\rbrack}$

[0070] and$\frac{\left( {n - 1} \right)s^{2}}{\sigma^{2}} \sim {\chi_{n - 1}^{2}.}$

[0071] So,$T^{*} = {\frac{y - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}} = {\frac{\frac{y - \overset{\_}{y}}{\sigma \sqrt{1 + \frac{1}{n}}}}{\sqrt{\frac{\left( {n - 1} \right)\frac{s^{2}}{\sigma^{2}}}{n - 1}}} \sim \frac{N\left\lbrack {0,1} \right\rbrack}{\sqrt{\frac{\chi_{n - 1}^{2}}{n - 1}}} \sim {t_{n - 1}.}}}$

[0072] Then,${\hat{p} = {P\left( {T^{*} \leq \frac{k_{L} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}} \right)}},{{{with}\quad T^{*}} \sim {t_{n - 1}.}}$

[0073] It is noted that the test statistic is the cumulativedistribution function, or cdf, value of a t statistic at some specificvalue, but it is not a t statistic itself.

[0074] The right-sided and two-sided cases may follow analogously. Forthe two-sided specification case, the test statistic is:$\hat{p} = {1 - {P\left( {\frac{k_{L} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}} \leq T^{*} \leq \frac{k_{U} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}} \right)}}$

[0075] while for the right-sided specification case, the test statisticis:$\hat{p} = {P{\left( {T^{*} \geq \frac{k_{U} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}} \right).}}$

[0076] In both cases, k_(U) represents the upper specification limit.The sample statistic {circumflex over (p)} may be the foundation onwhich to build a new test procedure.

[0077] 2.2 The Decision Rule

[0078] A goal of the present invention is to provide a sample statisticthat can truly estimate the process defect rate, not just someartificial statistic constructed via distribution theory, while notrequiring relinquishing power as compared to the ANSI test procedure. Aprimary benefit can be a better understanding of the ramifications onproduct quality of a particular numerical value for the samplestatistic. The hypothesis testing procedure chooses between the AQL andKDR values as to which is the more appropriate description of theunderlying process defect rate. The critical value, K, for thehypothesis test falls between the AQL and KDR, where the value for Kalso can provide information concerning sampling plan precision (inmeaningful terms) to the user. Specifically, the user can know thatcertain process defect rates above the AQL will likely beindistinguishable from the AQL.

[0079] Overall, the hypothesis test is right-tailed, with the statement“the process defect rate equals the AQL” being rejected whenever thesample statistic exceeds the critical value. Whether the actualspecifications for the response variable themselves are one-sided ortwo-sided can be immaterial regarding this decision rule.

[0080] The value of K is derived from the null distribution (i.e.assuming that p=AQL) of the test statistic. The right-tailed region ofthis distribution having an area of α provides the appropriate value forK in order to obtain a hypothesis test at an α level of significance. Toattain a specific power for the test, the alternate distribution of thetest statistic, evaluated at p=KDR, is used.

[0081] As stated above, either the KDR or the sample size may be fixedin advance. Given the KDR, the power determines the sample size, whichin turn is used with the AQL and α to compute K. Alternately, given thesample size, K can be computed directly from the AQL and α. In thiscase, calculating the KDR for a specific power is useful in determiningthe adequacy of the sampling plan, but the value of the KDR may have nodirect bearing on the decision rule.

[0082] 2.3 The Sampling Distribution of {circumflex over (p)}

[0083] To incorporate the sample statistic {circumflex over (p)} into aformal hypothesis test, the sampling distribution for {circumflex over(p)} is obtained. Unfortunately, the mathematics may become complex.Assuming a left-sided specification case, the cumulative distributionfunction, G, for {circumflex over (p)} is:${{G(w)} = {\int_{0}^{w}{{g\quad\left( {\int_{- \infty}^{\frac{k_{L} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}}{{f(t)}\quad {t}}} \right)}{z}}}},$

[0084] where g(z) is the probability density function for {circumflexover (p)} and f(t) is the probability density function of a tdistribution having n-1 degrees of freedom. However, the form of thefunction g(z) is generally unknown.

[0085] Without an expression (closed-form or otherwise) to define thesampling distribution for {circumflex over (p)}, Monte Carlo simulationmay be employed as one method of circumventing the complex distributiontheory. There is, however, one assumption that should be met, as willnow be described.

[0086] Setting the process defect rate to some constant value has theeffect of fixing the tailed area(s) under a normal curve to thisconstant value. As the normal distribution is a two-parameter (μ, σ²)distribution, there are an infinite number of (μ,σ²) pairs that willsatisfy this tailed area criterion. However, if μ is also fixed, then σ²can be solved for explicitly, and vice versa. A result from a previousMonte Carlo simulation study is that the sampling distribution for{circumflex over (p)} is invariant to the selection of the (μ, σ²) pair.For the sake of simplicity, σ² can therefore be arbitrarily set to 1,with μ computed directly from the true process defect rate. The randomnormal variates used in the Monte Carlo simulation are then fullydefined.

[0087] Monte Carlo simulation produces an empirical samplingdistribution for {circumflex over (p)} that is a function of both thesample size and the true process defect rate, p. The 100(1−α)^(th)percentile of the empirical cumulative distribution function, or ecdf,for {circumflex over (p)} can be used for determining K.

[0088] To illustrate the sampling distribution of {circumflex over (p)}graphically, consider two process defect rates (0.01 and 0.10) alongwith three sample sizes (5, 30, and 100). FIGS. 6A through 6F representhistograms of the Monte Carlo replicates (10,000 for each case) andindicate the shape of the probability distribution function for{circumflex over (p)}. It appears as though smaller sample sizes exhibitmore right-skewedness than do larger sample sizes. It also appears thatas the process defect rate becomes larger (and fixing the sample size),symmetry and normality are more closely adhered. Overall, the suggestionis that the sampling distribution for {circumflex over (p)} is unimodal.Given that {circumflex over (p)}≧0, many continuous distributions, suchas the Gamma and Weibull, could be investigated as to their potentialuse as approximations. Since, 0≦{circumflex over (p)}<1 is the realdomain for {circumflex over (p)}, a logical choice would be a Betadistribution. Further discussion concerning the modeling of the samplingdistribution of {circumflex over (p)} will be presented in Section2.7.2.

[0089] 2.4 The Bias of {circumflex over (p)}

[0090] Distribution theory suggests that {circumflex over (p)} estimatesp, but Monte Carlo simulation reveals an interesting fact: {circumflexover (p)} is a biased estimator of p. This fact is illustrated in bothTable 2A and Table 2B, where the mean of the Monte Carlo replicates forthe three illustrative sample sizes are all above the true value for p(for either p=0.01 or p=0.10). These outputs also show that the meansapproach the true value of p as the sample size increases. This meansthat {circumflex over (p)} is an asymptotically unbiased estimator of p.TABLE 2A Descriptive Statistics for Monte Carlo Replicates when p =0.01. Variable N Mean Median TrMean StDev SE Mean n = 5 10000 0.053670.04165 0.04952 0.04648 0.00046 n = 30 10000 0.01685 0.01422 0.015890.01196 0.00012 n = 100 10000 0.01196 0.01107 0.01165 0.00559 0.00006Variable Minimum Maximum Q1 Q3 n = 5 0.00001 0.31664 0.01726 0.07769 n =30 0.00001 0.09197 0.00795 0.02305 n = 100 0.00078 0.04245 0.007810.01518

[0091] TABLE 2B Descriptive Statistics for Monte Carlo Replicates when p= 0.10. Variable N Mean Median TrMean StDev SE Mean n = 5 10000 0.150660.13736 0.14530 0.09877 0.00099 n = 30 10000 0.10936 0.10614 0.108200.04161 0.00042 n = 100 10000 0.10272 0.10190 0.10236 0.02331 0.00023Variable Minimum Maximum Q1 Q3 n = 5 0.00002 0.80619 0.07248 0.21287 n =30 0.00779 0.26957 0.07905 0.13589 n = 100 0.03693 0.19259 0.086280.11825

[0092] The bias is positive, meaning that on average, {circumflex over(p)} will estimate the process defect rate as being slightly higher thanit truly is, with the quantification of slight being a function of thesample size and, to a lesser degree, the true process defect rate.Therefore, {circumflex over (p)} may be construed as being aconservative estimate of p. FIG. 7 shows the bias surface across boththe sample size and the true process defect rate (here, p=AQL). In FIGS.8A and 8B the bias trend is illustrated across various sample sizes fora fixed process defect rate.

[0093] In terms of the hypothesis testing framework, there do not appearto be any procedural difficulties stemming from the bias of {circumflexover (p)}. The false alarm rate and power can define the appropriatetailed regions of the null and alternate sampling distributions,respectively, and a sampling plan can be constructed. When performingthe data analysis, however, an estimate of the observed process defectrate should be available. In this light, improving {circumflex over (p)}is desirable, especially when dealing with small sample size scenarios.

[0094] 2.5 Bias Correction Regression

[0095] For a fixed sample size, the relationship between the true valuefor p and the expected value for {circumflex over (p)} will bedescribed. It is determined that the relationship betweenlog10(E_(n,p)[{circumflex over (p)}]) and log10(p) can be expressed witha cubic polynomial linear model. FIG. 9A illustrates this relationshipfor a sample size of n=5. The r-square is roughly 100%, with all threemodel terms being significant. Hence, there may be no need foroverfitting. This high degree of fit may be attained for all samplesizes.

[0096] In the hypothesis testing framework, the true value for p isassumed to be either the AQL or the KDR when determining either the nulldistribution or the alternate distribution. Therefore, the assumed truevalue for p can be used as the independent variable in the biascorrection regression, and the expected value for {circumflex over (p)}can be predicted. The sample statistic$\frac{p}{E_{n,p}\left\lbrack \hat{p} \right\rbrack}\hat{p}$

[0097] then becomes an unbiased test statistic, where p=AQL when dealingwith the null distribution and p=KDR when dealing with the alternatedistribution.

[0098] A different problem presents itself when the post-sampling dataanalysis is performed. When estimating the process defect rate, the biascorrection regression is reversed. Since p is unknown, it may not beused as an independent variable. Using the sample statistic {circumflexover (p)} as the independent variable E_(n,p)[{circumflex over (p)}],the true value for p can be estimated. FIG. 9B shows that a cubicpolynomial linear model (again with dual log10 transformations) performswell here as well.

[0099] There may be one inherent difficulty with the second biascorrection regression technique: the distribution of {circumflex over(p)} is not symmetric, thus replacing E_(n,p)[{circumflex over (p)}] by{circumflex over (p)} may not eliminate the bias, only help to reduceit.

[0100] 2.6 The Percentile Grid

[0101] The Monte Carlo simulation study to estimate the samplingdistribution for {circumflex over (p)} involves a rather extensive setof (n, p) pairs. Specifically, the sample sizes included in the studyare n=3,5(5)400(10)1000(200)2400,2500. The process defect rates includedin the study are p=0.0005, 0.001, 0.004, 0.0065, 0.01(0.01)0.15. Thesesample sizes and process defect rates are fully crossed, with 10,000Monte Carlo replicates of {circumflex over (p)} computed for each (n, p)pair. This large number of replicates is compressed into a set ofpercentiles for each (n, p) pair, using these percentiles:

[0102]{0.0001,0.0005,0.001,0.005,0.01(0.01)0.99,0.995,0.999,0.9995,0.9999}

[0103] These uncorrected sets of percentiles are then filtered throughan appropriate (a function of the sample size) bias correctionregression equation to produce the final sets of corrected percentiles,called the percentile grid because each row represents an (n, p) pairand each column a particular percentile.

[0104] A large number of (n, p) pairs were used for two primary reasons.First, the capturing of subtle curvilinear trends is desired, especiallysince the problem at hand mostly involves the extremes, or tails, ofsampling distributions. Second, with a fine resolution grid, a linearinterpolation procedure should provide an acceptable computationalmethod for addressing any value for n, p, or percentile not representedexplicitly by the grid. Other smoothing procedures may work as well, butthe trends being smoothed generally are monotone and the smoothingprocedure may need to account for this. As a counterexample, localquadratic fitting may not guarantee a monotone fit.

[0105] Regarding the linear interpolation procedure, the problem isessentially three-dimensional. There is the sample size, the processdefect rate, and the appropriate percentile (used to represent eitherthe false alarm rate or power). The linear interpolation procedure usedin the Excel workbooks described below incorporates a nearest neighborweighted average computation. To accomplish this, a 2³ factorial cube ofupper and lower grid bounds is determined to surround the point ofinterest. Each dimensional range is scaled to 1, then a Euclideandistance is computed for each of the cube corner points to the point ofinterest. Higher weights should be given to the smaller Euclideandistances. This is accomplished through the use of a harmonic meanexpression of the distances (which includes a constraint to have theweights sum to one). Finally, a weighted average is calculated.

[0106] 2.7 Distributional Approximations

[0107] The percentile grid method of estimating the samplingdistribution of {circumflex over (p)} may be considered an inelegant,brute force method. However, it may be generally desirable to have anaccurate representation over an elegant representation (if the cost isaccuracy). Several different methods may be used to include more elegantdistributional representations of the various items of interest. Thesemethods are summarized in the next two sections.

[0108] 2.7.1 Modeling the Critical Value and the Power

[0109] A consideration was made of modeling the critical value, K, as anonlinear function of the sample size, the AQL and the false alarm rate,and to model the power of the test as a function of the sample size, theAQL and the KDR. From a modeling standpoint, both regression models werehighly successful. The general regression model for the critical valuewas found to be

K˜β ₀ exp{β₁ n ^(βis 2)}+β₃ {square root}{square root over (n)}+β ₄ n,

[0110] with the last two terms helping to stabilize the asymptotictrend. Because the power of the test is bounded between 0 and 1, thereis a definite need to use a model that has a horizontal asymptote at apower equal to one. This leads to the statistical model for the power ofthe test being given by${power} \sim {\frac{1}{1 + {\gamma_{0}\exp \left\{ {{- \gamma_{1}}n^{\gamma_{2}}} \right\}}}.}$

[0111] Given the (AQL, KDR, α) triple, the sample size is found via${n = \left\lbrack {{- \frac{1}{\gamma_{1}}}{\ln \left( \frac{1 - {power}}{\gamma_{0}{power}} \right)}} \right\rbrack^{1/\gamma_{2}}},$

[0112] where γ₀, γ₁, γ₂˜f(AQL, KDR, α), with the function f representingthe dependence of the coefficients on the values for AQL, KDR, and α.Once the sample size is found, the critical value, K, is computed via

K=β ₀ exp{−β₁ n ^(β) ^(₂) }+β₃ {square root}{square root over (n)}+β ₄n, where β₀, β₁, β₂, β₃, β₄˜g(AQL, α),

[0113] with the function g representing the dependence of thecoefficients on the values for AQL and α.

[0114] There may be two major difficulties with this approach. First,the false alarm generally rate adds another dimension to the modelingproblem. Second, a different model may be needed for each potentialvalue of KDR−AQL. Together, these may lead to an enormous reference gridof coefficients. A solution to the problem may be to find a unifiedmodel for the critical value, and another unified model for the power.This means that the coefficients themselves are modeled as functions ofthe sample size, the AQL and the false alarm rate (in the critical valuecase), or the sample size, the AQL, the false alarm rate and the KDR (inthe power case). A unified model for the critical value was developed,but a unified power model was not developed. In any event, the noiseproduced through the estimation process of a unified model seemed tosuggest that there would be some loss in accuracy. Hence, a differentapproach was warranted.

[0115] 2.7.2 Modeling with Known Continuous Distributions

[0116] As stated in Section 2.3, the sampling distribution of{circumflex over (p)} for a given (n, p) pair can be modeled with acontinuous distribution. Even better may be the development of a unifiedmodel, where the coefficients themselves are a function of both n and p.

[0117] The two-parameter beta distribution is a logical choice formodeling the sampling distribution of {circumflex over (p)} because0≦{circumflex over (p)}≦1. However, distributional analysis over several(n, p) pairs found that both the two-parameter Weibull distribution andthe two-parameter Gamma distribution generally fit the data better thandoes the beta distribution. Even so, there are mixed results; sometimesthe fit is very good, but other times the tails are not wellrepresented. This is important because, due to the magnitude of both thefalse alarm rate and of the power, tail precision generally needs to bemaintained.

[0118] A three-parameter distribution is suggested as the minimallyparameterized distribution that will adequately describe the samplingdistribution of {circumflex over (p)}. The three-parameter Weibulldistribution and the generalized three-parameter beta distribution alsomay be considered. These distributions may include a true thirdparameter to aid in shape description, not just a “phase shift”parameter that moves the domain, but does not impact on the shape.

[0119] Accordingly, the percentile grid presently provides a moreaccurate estimation algorithm than other continuous distributionrepresentations.

[0120] 2.8 The QL Test

[0121] To summarize Sections 2.1-2.7.2, the test statistic used todiscriminate between the AQL and the KDR is defined to be${{\hat{QL}}_{0} = {\frac{AQL}{{\hat{E}}_{{AQL},n}\left\lbrack \hat{p} \right\rbrack}\hat{p}}},$

[0122] where {circumflex over (p)} is as defined in Section 2.1, andÊ_(AQL,n)[{circumflex over (p)}] is the predicted expected value for{circumflex over (p)} as computed via the bias correction regressionequation (conditioned on n) outlined in Section 2.5 under the assumptionthat p=AQL.

[0123] The decision rule is to reject the null hypothesis that theprocess defect rate equals the AQL provided that {circumflex over(Q)}L₀>K. The value of K is determined via the percentile grid, withlinear interpolation used whenever necessary.

[0124] The actual quality level point estimate is defined as:

{circumflex over (Q)}L=10^(θ) ^(₀) ^(+θ) ^(₁)^(log10({circumflex over (p)})+θ) ^(₂) ^((log10({circumflex over (p)})))² ^(+θ) ^(₃) ^((log10({circumflex over (p)}))) ³ ,

[0125] where θ is a function of the sample size. As stated before, thisis a conservative estimate of the process defect rate since, on average,the true process defect rate is slightly less than this estimated value.

[0126] 3. Detailed Description of User Interface

[0127] The user interface preferably includes three customized MicrosoftExcel workbooks, as follows:

[0128] Attributes_SP.xls, creates sampling plans for attributes data.

[0129] Variables_SP.xls, creates sampling plans for variables data.

[0130] Variables_DA.xls, analyzes variables data via the QL test.

[0131] However, other spreadsheet programs or other applicationsprograms or custom programs maybe used.

[0132] Attributes data, being dichotomous in nature, are evaluated withrespect to binomial distribution theory (assuming “infinite” lot sizes).This is consistent with the widely accepted methodology for producingsampling plans.

[0133] Variables data preferably is analyzed via the QL test, as wasdescribed above. The usual normal distribution framework still applies,but an improvement in power and a vast improvement in flexibility may beprovided over the ANSI standard approach.

[0134] An overview of the hypothesis test framework as it pertains toeither attributes data or variables data first will be presented.

[0135] 3.1 Attributes Data

[0136] The hypothesis test is of the form

H₀: p=AQL,

H₁: p>AQL,

[0137] with the lot being rejected if X>c. Following the binomialdistribution for large lot sizes, the false alarm rate, or type I error,is determined from the relation$\alpha = {\sum\limits_{x = {c + 1}}^{n}\quad {\begin{pmatrix}n \\x\end{pmatrix}{{{AQL}^{x}\left( {1 - {AQL}} \right)}^{n - x}.}}}$

[0138] With c fixed and β set for a prescribed KDR, the sample size isdetermined, regardless of the AQL, by solving the following expressionfor n: $\beta = {\sum\limits_{x = 0}^{c}\quad {\begin{pmatrix}n \\x\end{pmatrix}{{{KDR}^{x}\left( {1 - {KDR}} \right)}^{n - x}.}}}$

[0139] Without a closed-form expression, the sample size can be computedin an iterative manner.

[0140] 3.2 Variables Data

[0141] The hypothesis test is of the form

H₀: p=AQL,

H₁: p>AQL.

[0142] For a left-sided specification case, the test statistic for theQL test is${\hat{p} = {P\left( {T^{*} \leq \frac{k_{L} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}} \right)}},{{{with}\quad T^{*}} \sim {t_{n - 1}.}}$

[0143] It is noted that the test statistic is the cdf (cumulativedistribution function) value of a t statistic at some specific value,but it is not a t statistic itself.

[0144] For the two-sided specification case the test statistic is${\hat{p} = {1 - {P\left( {\frac{k_{L} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}} \leq T^{*} \leq \frac{k_{U} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}} \right)}}},{{{with}\quad T^{*}} \sim {t_{n - 1}.}}$

[0145] Finally, for the right-sided specification case the teststatistic is${\hat{p} = {P\left( {T^{*} \geq \frac{k_{U} - \overset{\_}{y}}{s\sqrt{1 + \frac{1}{n}}}} \right)}},{{{with}\quad T^{*}} \sim {t_{n - 1}.}}$

[0146] The decision rule is one-sided regardless of whether thespecifications are left-sided, right-sided, or two-sided. Namely, thelot is rejected if {circumflex over (p)}>K.

[0147] 4. Overview of the Microsoft Excel Workbooks

[0148] As stated above, three (3) Microsoft Excel workbooks may beprovided:

[0149] Attributes_SP.xls, creates sampling plans for attributes data.

[0150] Variables_SP.xls, creates sampling plans for variables data.

[0151] Variables_DA.xls, analyzes variables data via the QL test.

[0152] There need be no Attributes_DA.xls file because there generallyis no need for a data analysis spreadsheet when the attributes decisionrule is simply whether or not the number of observed defectives exceedssome critical value.

[0153] Section 5 contains detailed illustration of how various scenariosare addressed with these workbooks.

[0154] 5. Detailed Description for Using the Microsoft Excel Workbooks

[0155] 5.1 Attributes_SP.xls

[0156]FIG. 10 shows the layout of the worksheet Attributes_SP.xls. Thecontrol panel is set up in columns B through F. Three (3) charts appearin columns G through P:

[0157] OC Curve. (Upper Left) A plot of the probability of accepting alot (vertical axis) against the process defect rate (horizontal axis).

[0158] AOQ Curve. (Upper Right) A plot of the outgoing defect rate(vertical axis) against the process defect rate (horizontal axis).

[0159] Maintaining the Key Defect Rate. (Bottom) A plot of the a error(vertical axis) against the critical value (horizontal axis). The datalabels represent the sample size required to maintain fixed values forboth the KDR and the power at KDR.

[0160] As seen in FIG. 10, only eight (8) cells can be changed. SeeTable 3: TABLE 3 Editable Cell Purpose D4 User specified n D5 Userspecified c D6 AQL D7 β error (becomes target when maintaining the KDR)D16 User specified KDR E29 Maximum defect rate for OC and AOQ curves E31Maximum critical value for Maintaining the KDR graph E32 Nominal αerror, required when maintaining the KDR

[0161] A user has two options available for the construction of aparticular sampling plan:

[0162] 1. Specify the sample size, critical cutoff value, AQL, and βerror, then evaluate this sampling plan by viewing the resulting α errorand KDR. The “OC curve” and “AOQ curve” can then be used to get a moregeneral perspective of the sampling plan and determine if it ispractically feasible and still functional.

[0163] 2. Specify the KDR, β, error, AQL, α error (target value), andcritical value, then determine the required sample size. It isrecommended that initially the critical value be set to 0, with the“Maintaining the KDR” graph providing its own evaluation of the samplesize and critical value pair that keeps the true a error as close to thetargeted (or nominal) value as possible.

[0164] The procedure for each of these two cases is outlined in thefollowing two sections. To alleviate intermittent calculations frombecoming a nuisance, the workbook has the manual calculation optionselected. Therefore, the F9 key is used to recalculate the worksheetwhenever necessary.

[0165] 5.1.1 Attributes Data Sampling Plan: Unspecified KDR Case

[0166] The first scenario considered is the evaluation of auser-specified sampling plan. The sample size is entered in cell D4,with the critical value in cell D5, the AQL in cell D6, and the β errorin cell D7. The resulting α error is displayed in cell D9, with thecomputed KDR found in cell D10.

EXAMPLE 1

[0167] Suppose that a user wants to evaluate a sampling plan involving asample size of n=50 items with a c=0 acceptance criterion. The AQL is tobe 0.65% and the test should provide 90% power.

[0168] Results: Example 1

[0169] Entering the four given values into cells D4:D7 as outlinedabove, it is seen that the α error is 27.82%, with the KDR being 4.50%.A maximum defect rate of 6% (slightly above KDR) is entered in cell E29in order to provide a decent horizontal axis scale. The spreadsheet isrecalculated using the F9 key. FIG. 11 illustrates this spreadsheet,which now shows an AOQL of 0.73%. The Maintaining the KDR graphindicates a setup problem due to the empty D16, B31, and E31 cells.However, this scenario does not attempt to control the KDR, so thiserror message should be ignored.

[0170] 5.1.2 Attributes Data Sampling Plan: Specified KDR Case

[0171] The second scenario considered is to solve for the sampling planparameters (n, c) in order to maintain a specific KDR at a given powerunder the conditions of a prescribed AQL and associated α error. The AQLis entered in cell D6, with the β error in cell D7 and the KDR in cellD16. In addition, the targeted α error is entered in cell E31, with amaximum critical value in cell B31 and an initial critical value in cellD5.

EXAMPLE 2

[0172] Suppose that it is desired to maintain a KDR of 3% with 95%probability. The false alarm rate is to be α=10% when the process is atAQL=1%. It is desired to compute the sampling plan parameters: namelythe sample size, n, and critical cutoff, c.

[0173] Results: Example 2

[0174] Ignore cell D4, the sample size entry. Arbitrarily set c=0, thusentering “0” in cell D5. Then, enter “1” (AQL=1%) in cell D6, followedby entering “5” (β=5%) in cell D7. Next, enter “3” (KDR=3%) in cell D16.The targeted α error is entered as “10” in cell E31, with a maximumcritical value of “15” in cell B31. Recalculate the spreadsheet usingthe F9 key. From this, it is seen in FIG. 12 (viewing cell D17) thatn=99 is needed. However, under these conditions, the false alarm rate ishighly unacceptable (see cell D23), as α=63%. Lowering n would lower αbut increase the KDR. Increasing n will increase α. Therefore, c shouldbe increased.

[0175] To avoid forcing the user into performing an iterative search,the Maintaining the KDR chart is consulted. It is seen that therecommended sampling plan has parameters n=392 and c=6.

[0176] It will be understood that in FIG. 12, the OC curve and AOQ curveindicate setup problems, which are also signaled with warning lights forcells D4 and E29. Entering a “392” into cell D4 and a “6” into cell E29,along with a maximum defect rate (here use 6%, so enter a “6”) in cellE29, finalizes the procedure. Lowering the maximum critical value to “7”in cell B31 would reduce the computation time needed for therecalculation (the F9 key), although this is not required. FIG. 13displays the final spreadsheet layout for this example. Here it is seenthat the AOQL is 0.97%.

[0177] While a sample size of 392 may appear large, examine theconstraints of the problem. The difference between AQL and KDR is only2%, and higher precision generally means higher sample size. Inaddition, with both AQL and KDR being small numerically, the scenario isalso affected by the rare events identification problem. This situationalso may require larger sample sizes.

[0178] 5.2 Variables_SP.xls

[0179]FIG. 14 shows the layout of the variables data sampling planworksheet Variables_SP.xls. The control panel is set up in columns Bthrough J. Two (2) charts appear in columns L through Q:

[0180] OC Curve. (Upper Left) A plot of the probability of accepting alot (vertical axis) against the process defect rate (horizontal axis).

[0181] AOQ Curve. (Upper Right) A plot of the outgoing defect rate(vertical axis) against the process defect rate (horizontal axis).

[0182] As seen in FIG. 14, only seven (7) cells can be changed. SeeTable 4. TABLE 4 Editable Cell Purpose F4 AQL F5 α error F6 power E9User specified n G9 User specified KDR C18 Maximum defect rate for OCand AOQ curves C19 Case to plot (1 or 2) 1 = prescribe n 2 = prescribeKDR

[0183] A user has two options available when constructing a particularvariables data sampling plan:

[0184] 1. Specify the KDR, β error, AQL, and α error, then determine therequired sample size. The critical value is determined as a function ofAQL and α.

[0185] 2. Specify the sample size, AQL, and α error, and power, thenevaluate this sampling plan by viewing the resulting KDR. Again, thecritical value is determined as a function of AQL and α.

[0186] The OC curve and AOQ curve can be used to critique the resultingsampling plan for practical feasibility and functionality. The procedurefor each of these two cases is outlined in the following two sections.Unlike Attributes_SP.xls, the workbook Variables_SP.xls does not havethe manual calculation option selected, so the F9 key is not neededhere.

[0187] 5.2.1 Variables Data Sampling Plan: Unspecified KDR Case

[0188] Consider now the evaluation of a user-specified variables datasampling plan. The AQL is entered in cell F4, with the a error in cellF5 and the power in cell F6. The sample size is entered in cell G9. Forplotting purposes, a maximum defect rate is entered in cell C18 and thecase identifier “2” is entered in cell C19.

EXAMPLE 3

[0189] Suppose that the user wants to evaluate a sampling plan involvinga sample size of n=50 items. The AQL is to be 0.65% with a false alarmrate of α=5%, and the test should provide 90% power.

[0190] After entering a “0.65” in cell F4, a “5” in cell F5, a “90” incell F6, and a “50” in cell G9, the resulting KDR of 4.025% is displayedin cell G11. Also, the critical value of 2.263% is displayed in cellG12. With the KDR being 4.025%, a maximum defect rate of 5% seemsreasonable for the two plots. Thus, a “5” is entered in cell C18, withthe case identifier “2” entered in cell C19. FIG. 15 shows the finalspreadsheet layout for example 3, where it is seen that the AOQL is1.02%.

[0191] 5.2.2 Variables Data Sampling Plan: Specified KDR Case

[0192] Next, consider the construction of a variables data sampling planwhen the KDR is fixed in advance. Also given is the power at the KDR,the AQL, and the α error. The AQL is entered in cell F4, with the αerror in cell F5 and the power in cell F6. The KDR is entered in cellE9. For plotting purposes, a maximum defect rate is entered in cell C18and the case identifier “1” is entered in cell C19.

EXAMPLE 4

[0193] Suppose that it is desired to obtain 95% power at KDR 3%, withthe false alarm rate being α=5% when the process is at AQL=1%. It isdesired to compute the sampling plan parameters: namely the sample size,n, and critical value, K.

[0194] Results: Example 4

[0195] After entering a “1” in cell F4, a “5” in cell F5, a “95” in cellF6, and a “3” in cell E9, the resulting sample size of 174 is displayedin cell E11. Also, the critical value of 1.86% is displayed in cell E12.With the KDR being 3%, a maximum defect rate of 5% seems reasonable forthe two plots. Thus, a “5” is entered in cell C18, with the caseidentifier “1” entered in cell C19. FIG. 16 shows the final spreadsheetlayout for example 4, where the AOQL is 1.05%.

[0196] It is the ease at which Example 4 is performed that illustratesthe gains in flexibility that the QL test procedure has over the ANSIstandard approach. The next section deals with the analysis of variablesdata when using the QL test procedure.

[0197] 5.3 Variables_DA.xls

[0198] The general spreadsheet layout of the variables data analysisworkbook Variables_DA.xls is shown in FIG. 17. The data analysis summaryis self-contained within one graph. Annotation is provided to report theparticulars of the hypothesis test, the point estimate for the qualitylevel, and an explicit conclusion.

[0199] As seen in FIG. 17, there are eight (8) editable cells devoted tothe setup of the problem, and one long editable data field. See Table 5.TABLE 5 Editable Cell Purpose Editable Cell Purpose C4 Response name G4AQL C5 Lower specification G5 KDR limit C6 Upper specification G6 K, thecritical value limit C11:C2510 Raw data entry G7 α G8 power

[0200] While the workbook can handle response variables with either aone-sided or two-sided specification range, only the left-sidedspecification range is illustrated here.

EXAMPLE 5

[0201] Suppose that the response variable is separation force (ft.lbs.),and the lower specification limit is set at 8 ft.lbs. Assume that theconditions of Example 4 apply here: 95% power at KDR=3%, α=5% at AQL=1%.The sample size is n=174 and the critical value is K=1.86%.

[0202] Results: Example 5

[0203] To begin the analysis, enter the response name “Separation Force”into cell C4, and the lower specification limit equaling “8” in cell C5.The upper specification limit cell C6 is left blank. Then, thehypothesis test information is entered: a “1” in cell G4, a “3” in cellG5, a “1.86” in cell G6, a “5” in cell G7, and a “95” in cell G8.Finally, the data is placed in cells C11:C184. It is seen in FIG. 18that the QL estimate is 0.84% and that the conclusion is to accept thenull hypothesis that the process defect rate is at AQL.

[0204] The normal curve displayed on the chart uses the sample mean andsample standard deviation as its parameters, with the mean beingoverlaid onto the curve. The specification limit(s) is(are) highlightedas well. However, it is again stressed that the QL estimate is not justthe area under the curve beyond the specification limit(s): this areawould not account for the random variability of either sample statisticsused to generate the normal curve. This curve does, however, offer asimple graphical guide for ancillary purposes.

[0205] 6. Conclusion

[0206] As shown, the QL test procedure for acceptance sampling canimprove the ANSI standard procedure by allowing a tremendous gain inflexibility regarding the specification of the sampling plan parameters.By eliminating the need to use the vast quantity of ANSI tables andgraphs and replacing them instead with an Excel workbook, improvedconstruction of sampling plans can be provided. A less intimidating,less confusing, and more convenient tool to aid the non-statisticallyoriented user in constructing and analyzing acceptance sampling plansfor variables data can be provided by the QL test.

[0207] In terms of computational issues, the results presently aremixed. It is true that the non-central t distribution and its complexityin obtaining tailed areas and critical values has been avoided. However,a large percentile grid may need to be stored, and the linearinterpolation procedures can be rather involved. The QL test proceduredoes have the potential of becoming very computationally efficient. Inparticular, a unified model for the sampling distribution of {circumflexover (p)} may be developed. Such a model could eliminate the need of anyinterpolation algorithms and could make the sampling plan constructioncomputations closed-formed.

[0208] It also has been shown that the three (3) Microsoft Excelworkbook files can facilitate the construction of sampling plans foreither attributes data or variables data. It also has been shown thatthe flexibility of the QL test procedure can be a vast improvement oversifting through a book of standards to obtain a sampling plan that mayor may not be adequate. Computer resources have made obsolete the“one-size-fits-all” philosophy. With the growing trend towardstatistical training in industry, the notion of using tables and chartsfor the sole purpose of keeping things parsimonious, at the expense ofaccuracy or correctness, is also outdated. Finally, it is recommendedthat since sampling plans can be constructed with the KDR specified inadvance, it may be in the best interest of the user to determine andincorporate a value for the KDR via practical considerations. Thisaction can protect the process from undetected drifts into anunacceptably high defect rate region.

[0209] In the drawings and specification, there have been disclosedtypical preferred embodiments of the invention and, although specificterms are employed, they are used in a generic and descriptive senseonly and not for purposes of limitation, the scope of the inventionbeing set forth in the following claims.

What is claimed is:
 1. A computerized method of constructing a samplingplan for items that are manufactured, comprising the steps of: inputtinginto a computer, a desired Acceptable Quality Limit (AQL), a desired KeyDefect Rate (KDR), a desired power of the sampling plan for the itemsthat are manufactured and a desired false alarm rate for the samplingplan; and calculating in the computer, a required sample size to providethe desired AQL, the desired KDR, the desired power of the sampling planfor the items that are manufactured and the desired false alarm rate forthe sampling plan.
 2. A method according to claim 1 wherein thecalculating step further comprises the step of calculating in thecomputer a decision rule critical value based upon the required samplesize to provide the desired AQL, the desired KDR, the desired power ofthe sampling plan for the items that are manufactured and the desiredfalse alarm rate for the sampling plan.
 3. A method according to claim 2wherein the calculating step is followed by the step of graphicallydisplaying a relationship between sample size, acceptable number ofdefective items and false alarm rate, based upon the desired AQL, thedesired KDR and the desired power of the sampling plan for the itemsthat are manufactured.
 4. A method according to claim 2 wherein thecalculating step is followed by the steps of: sampling the items thatare manufactured at the required sample size to obtain samples; anddetermining the number of defective items in the samples.
 5. A methodaccording to claim 2 wherein the calculating step is followed by thesteps of: sampling the items that are manufactured at the requiredsample size to obtain samples; and measuring a response variable for theeach of the samples.
 6. A method according to claim 5 wherein themeasuring step is followed by the steps of: inputting into the computerthe measured response variable for each of the samples; and calculatingin the computer, an estimate of the quality level (QL) for the itemsthat are manufactured, based on the measured response variable for eachof the samples.
 7. A method according to claim 1 wherein the calculatingstep is preceded by the steps of: calculating a sampling distributionthat is variance invariant based on a normal distribution; formulating apercentile grid of sample size and a true process defect rate, based onestimated percentiles of a cumulative distribution of the samplingdistribution; formulating a bias corrected percentile grid of samplesize and the true process defect rate from the percentile grid; andstoring the bias corrected percentile grid in the computer.
 8. A methodaccording to claim 1 wherein the calculating step comprises the stepsof: computing a decision rule critical value from the AQL and the falsealarm rate, across a plurality of sample sizes, using a bias correctedpercentile grid of sample size and a true process defect rate; andevaluating the bias corrected percentile grid for the decision rulecritical value, to determine a sample size.
 9. A method according toclaim 6 wherein the step of calculating an estimate of QL comprises thesteps of: computing a bias correction coefficient; computing a QL teststatistic as a function of the bias correction coefficient and at leastone quantile from a cumulative distribution function of a central tdistribution with at least one argument that is a function of a samplemean, a sample standard deviation, the sample size and specificationlimits; and determining whether the QL test statistic is at least equalto the decision rule critical value.
 10. A method according to claim 5wherein the measuring step is followed by the steps of: inputting intothe computer the measured response variable for each of the samples; andcalculating in the computer, a point estimate of the number ofout-of-specification items that are manufactured, based on the measuredresponse variable for each of the samples.
 11. A computerized method ofconstructing a sampling plan for items that are manufactured, comprisingthe steps of: inputting into a computer, a desired sample size, adesired false alarm rate, a desired Acceptable Quality Limit (AQL) and adesired power of the sampling plan for the items that are manufactured;and calculating in the computer, a Key Defect Rate (KDR) that isproduced from the desired sample size, the desired false alarm rate, thedesired AQL and the desired power of the sampling plan for the itemsthat are manufactured.
 12. A method according to claim 11 wherein thecalculating step comprises the step of: computing a decision rulecritical value based on the desired AQL and the desired false alarm ratefor the desired sample size.
 13. A method according to claim 11 whereinthe calculating step is followed by the step of graphically displaying arelationship between acceptable number of defective items and falsealarm rate, based upon the desired AQL, the desired KDR and the desiredpower of the sampling plan for the items that are manufactured.
 14. Amethod according to claim 11 wherein the calculating step is followed bythe steps of: sampling the items that are manufactured at the desiredsample size to obtain samples; and determining the KDR of the items thatare manufactured from the samples.
 15. A method according to claim 11wherein the calculating step is followed by the steps of: sampling theitems that are manufactured at the desired sample size to obtainsamples; and measuring a response variable for the each of the samples.16. A method according to claim 15 wherein the measuring step isfollowed by the steps of: inputting into the computer the measuredresponse variable for each of the samples; and calculating in thecomputer, an estimate of the KDR for the items that are manufactured,based on the measured response variable for each of the samples.
 17. Amethod according to claim 11 wherein the calculating step is preceded bythe steps of: calculating a sampling distribution that is varianceinvariant based on a normal distribution; formulating a percentile gridof sample size and a true process defect rate, based on estimatedpercentiles of a cumulative distribution of the sampling distribution;formulating a bias corrected percentile grid of sample size and the trueprocess defect rate from the percentile grid; and storing the biascorrected percentile grid in the computer.
 18. A method according toclaim 12 wherein the calculating step comprises the steps of: computinga decision rule critical value from the AQL, the false alarm rate andthe desired sample size, using a bias corrected percentile grid ofsample size and a true process defect rate; and evaluating the biascorrected percentile grid for values that are larger than the AQL, withthe percentile being the desired power.
 19. A method according to claim15 wherein the measuring step is followed by the steps of: inputtinginto the computer the measured response variable for each of thesamples; and calculating in the computer, a point estimate of a processdefect rate for the items that are manufactured, based on the measuredresponse variable for each of the samples.
 20. A computer system forconstructing a sampling plan for items that are manufactured, the systemcomprising: means for inputting a desired Acceptable Quality Limit(AQL), a desired Key Defect Rate (KDR), a desired power of the samplingplan for the items that are manufactured and a desired false alarm ratefor the sampling plan; and means for calculating a required sample sizeto provide the desired AQL, the desired KDR, the desired power of thesampling plan for the items that are manufactured and the desired falsealarm rate for the sampling plan.
 21. A system according to claim 20wherein the calculating means further comprises means for calculating adecision rule critical value based upon the required sample size toprovide the desired AQL, the desired KDR, the desired power of thesampling plan for the items that are manufactured and the desired falsealarm rate for the sampling plan.
 22. A system according to claim 21further comprising: means for graphically displaying a relationshipbetween sample size, acceptable number of defective items and falsealarm rate, based upon the desired AQL, the desired KDR and the desiredpower of the sampling plan for the items that are manufactured.
 23. Asystem according to claim 21 further comprising: means for inputting ameasured response variable for each of a plurality of samples of theitems that are manufactured; and means for calculating an estimate ofthe quality level (QL) for the items that are manufactured, based on themeasured response variable for each of the samples.
 24. A systemaccording to claim 20 further comprising: means for calculating asampling distribution that is variance invariant based on a normaldistribution; means for formulating a percentile grid of sample size anda true process defect rate, based on estimated percentiles of acumulative distribution of the sampling distribution; means forformulating a bias corrected percentile grid of sample size and the trueprocess defect rate from the percentile grid; and means for storing thebias corrected percentile grid.
 25. A system according to claim 20wherein the means for calculating comprises: means for computing adecision rule critical value from the AQL and the false alarm rate,across a plurality of sample sizes, using a bias corrected percentilegrid of sample size and a true process defect rate; and means forevaluating the bias corrected percentile grid for the decision rulecritical value, to determine a sample size.
 26. A system according toclaim 23 wherein the means for calculating an estimate of QL comprises:means for computing a bias correction coefficient; means for computing aQL test statistic as a function of the bias correction coefficient andat least one quantile from a cumulative distribution function of acentral t distribution with at least one argument that is a function ofa sample mean, a sample standard deviation, the sample size andspecification limits; and means for determining whether the QL teststatistic is at least equal to the decision rule critical value.
 27. Asystem according to claim 23 further comprising: means for inputting themeasured response variable for each of a plurality of samples of theitems that are manufactured; and means for calculating a point estimateof the number of out-of-specification items that are manufactured, basedon the measured response variable for each of the samples.
 28. Acomputer system for constructing a sampling plan for items that aremanufactured, the system comprising: means for inputting a desiredsample size, a desired false alarm rate, a desired Acceptable QualityLimit (AQL) and a desired power of the sampling plan for the items thatare manufactured; and means for calculating a Key Defect Rate (KDR) thatis produced from the desired sample size, the desired false alarm rate,the desired AQL and the desired power of the sampling plan for the itemsthat are manufactured.
 29. A system according to claim 28 wherein themeans for calculating comprises: means for computing a decision rulecritical value based on the desired AQL and the desired false alarm ratefor the desired sample size.
 30. A system according to claim 28 furthercomprising: means for graphically displaying a relationship betweenacceptable number of defective items and false alarm rate, based uponthe desired AQL, the desired KDR and the desired power of the samplingplan for the items that are manufactured.
 31. A system according toclaim 28 further comprising: means for inputting a measured responsevariable for each of a plurality of samples of the items that aremanufactured; and means for calculating an estimate of the KDR for theitems that are manufactured, based on the measured response variable foreach of the samples.
 32. A system according to claim 28 furthercomprising: means for calculating a sampling distribution that isvariance invariant based on a normal distribution; means for formulatinga percentile grid of sample size and a true process defect rate, basedon estimated percentiles of a cumulative distribution of the samplingdistribution; means for formulating a bias corrected percentile grid ofsample size and the true process defect rate from the percentile grid;and means for storing the bias corrected percentile grid.
 33. A systemaccording to claim 29 wherein the means for calculating comprises: meansfor computing a decision rule critical value from the AQL, the falsealarm rate and the desired sample size, using a bias correctedpercentile grid of sample size and a true process defect rate; and meansfor evaluating the bias corrected percentile grid for values that arelarger than the AQL, with the percentile being the desired power.
 34. Asystem according to claim 28 further comprising: means for inputting ameasured response variable for each of a plurality of samples of theitems that are manufactured; and means for calculating a point estimateof a process defect rate for the items that are manufactured, based onthe measured response variable for each of the samples.
 35. A computerprogram product for constructing a sampling plan for items that aremanufactured, the computer program product comprising acomputer-readable storage medium having computer-readable program codemeans embodied in the medium, the computer-readable program code meanscomprising: computer-readable program code means for accepting a desiredAcceptable Quality Limit (AQL), a desired Key Defect Rate (KDR), adesired power of the sampling plan for the items that are manufacturedand a desired false alarm rate for the sampling plan; andcomputer-readable program code means for calculating a required samplesize to provide the desired AQL, the desired KDR, the desired power ofthe sampling plan for the items that are manufactured and the desiredfalse alarm rate for the sampling plan.
 36. A computer program productaccording to claim 35 wherein the computer-readable program codecalculating means further comprises means for calculating a decisionrule critical value based upon the required sample size to provide thedesired AQL, the desired KDR, the desired power of the sampling plan forthe items that are manufactured and the desired false alarm rate for thesampling plan.
 37. A computer program product according to claim 35further comprising: computer-readable program code means for graphicallydisplaying a relationship between sample size, acceptable number ofdefective items and false alarm rate, based upon the desired AQL, thedesired KDR and the desired power of the sampling plan for the itemsthat are manufactured.
 38. A computer program product according to claim35 further comprising: computer-readable program code means foraccepting a measured response variable for each of a plurality ofsamples of the items that are manufactured; and computer-readableprogram code means for calculating an estimate of the quality level (QL)for the items that are manufactured, based on the measured responsevariable for each of the samples.
 39. A computer program productaccording to claim 35 further comprising: computer-readable program codemeans for calculating a sampling distribution that is variance invariantbased on a normal distribution; computer-readable program code means forformulating a percentile grid of sample size and a true process defectrate, based on estimated percentiles of a cumulative distribution of thesampling distribution; computer-readable program code means forformulating a bias corrected percentile grid of sample size and the trueprocess defect rate from the percentile grid; and computer-readableprogram code means for storing the bias corrected percentile grid.
 40. Acomputer program product according to claim 35 wherein thecomputer-readable program code means for calculating comprises:computer-readable program code means for computing a decision rulecritical value from the AQL and the false alarm rate across a pluralityof sample sizes, using a bias corrected percentile grid of sample sizeand a true process defect rate; and computer-readable program code meansfor evaluating the bias corrected percentile grid for the decision rulecritical value, to determine a sample size.
 41. A computer programproduct according to claim 38 wherein the computer-readable program codemeans for calculating an estimate of QL comprises: computer-readableprogram code means for computing a bias correction coefficient;computer-readable program code means for computing a QL test statisticas a function of the bias correction coefficient and at least onequantile from a cumulative distribution function of a central tdistribution with at least one argument that is a function of a samplemean, a sample standard deviation, the sample size and specificationlimits; and computer-readable program code means for determining whetherthe QL test statistic is at least equal to the decision rule criticalvalue.
 42. A computer program product according to claim 38 furthercomprising: computer-readable program code means for accepting themeasured response variable for each of a plurality of samples of theitems that are manufactured; and computer-readable program code meansfor calculating a point estimate of the number of out-of-specificationitems that are manufactured, based on the measured response variable foreach of the samples.
 43. A computer program product for constructing asampling plan for items that are manufactured, the computer programproduct comprising a computer-readable storage medium havingcomputer-readable program code means embodied in the medium, thecomputer-readable program code means comprising: computer-readableprogram code means for accepting a desired sample size, a desired falsealarm rate, a desired Acceptable Quality Limit (AQL) and a desired powerof the sampling plan for the items that are manufactured; andcomputer-readable program code means for calculating a Key Defect Rate(KDR) that is produced from the desired sample size, the desired falsealarm rate, the desired AQL and the desired power of the sampling planfor the items that are manufactured.
 44. A computer program productaccording to claim 43 wherein the computer-readable program code meansfor calculating comprises: computer-readable program code means forcomputing a decision rule critical value based on the desired AQL andthe desired false alarm rate for the desired sample size.
 45. A computerprogram product according to claim 43 further comprising:computer-readable program code means for graphically displaying arelationship between acceptable number of defective items and falsealarm rate, based upon the desired AQL, the desired KDR and the desiredpower of the sampling plan for the items that are manufactured.
 46. Acomputer program product according to claim 43 further comprising:computer-readable program code means for accepting a measured responsevariable for each of a plurality of samples of the items that aremanufactured; and computer-readable program code means for calculatingan estimate of the KDR for the items that are manufactured, based on themeasured response variable for each of the samples.
 47. A computerprogram product according to claim 43 further comprising:computer-readable program code means for calculating a samplingdistribution that is variance invariant based on a normal distribution;computer-readable program code means for formulating a percentile gridof sample size and a true process defect rate, based on estimatedpercentiles of a cumulative distribution of the sampling distribution;computer-readable program code means for formulating a bias correctedpercentile grid of sample size and the true process defect rate from thepercentile grid; and computer-readable program code means for storingthe bias corrected percentile grid.
 48. A computer program productaccording to claim 44 wherein the computer-readable,program code meansfor calculating comprises: computer-readable program code means forcomputing a decision rule critical value from the AQL, the false alarmrate and the desired sample size, using a bias corrected percentile gridof sample size and a true process defect rate; and computer-readableprogram code means for evaluating the bias corrected percentile grid forvalues that are larger than the AQL, with the percentile being thedesired power.
 49. A computer program product according to claim 43further comprising: computer-readable program code means for inputting ameasured response variable for each of a plurality of samples of theitems that are manufactured; and computer-readable program code meansfor calculating a point estimate of a process defect rate for the itemsthat are manufactured, based on the measured response variable for eachof the samples.